As the population variance is not given, we have to use the T-Statistics:

.

P(T(df = 29) > To = − 1.733) = 0.047, thus

the for this (double-sided) test.

Therefore, we can not reject the null hypothesis at α = 0.05! The left and right white areas at the tails of the T(df=29) distribution depict graphically the probability of interest, which represents the strength of the evidence (in the data) against the Null hypothesis. In this case, the cumulative tail area is 0.094, which is larger than the initially set Type I error α = 0.05 and we can not reject the null hypothesis.

Examples

Cavendish Mean Density of the Earth

A number of famous early experiments of measuring physical constants have later been shown to be biased. In the 1700's Henry Cavendish measured the Mean density of the Earth. Formulate and test null and research hypotheses about these data regarding the now know exact mean-density value = 5.517. These sample statistics may be helpful

n = 23, sample mean = 5.483, sample SD = 0.1904

5.36

5.29

5.58

5.65

5.57

5.53

5.62

5.29

5.44

5.34

5.79

5.10

5.27

5.39

5.42

5.47

5.63

5.34

5.46

5.30

5.75

5.68

5.85

Hypothesis Testing Summary

Important parts of Hypothesis Test conclusions:

Decision (significance or no significance)

Parameter of Interest

Variable of Interest

Population under study

(optional but preferred) P-value

Parallels between Hypothesis Testing and Confidence Intervals

These are different methods for coping with the uncertainty about the true value of a parameter caused by the sampling variation in estimates.

Confidence Intervals: A fixed level of confidence is chosen. We determine a range of possible values for the parameter that are consistent with the data (at the chosen confidence level).

Hypothesis (Significance) testing: Only one possible value for the parameter, called the hypothesized value, is tested. We determine the strength of the evidence (confidence) provided by the data against the proposition that the hypothesized value is the true value.